A connected graph is double connected if its complement is also connected The following Ramsey type theorem is proved in this paper There exists a function h n de ned on the set of integers exceeding three such that every double connected graph on at least h n vertices must contain as an induced subgraph a special double connected graph which is either one of the following graphs or the complem...

A connected graph is double-connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), deened on the set of integers exceeding three, such that every double-connected graph on at least h(n) vertices must contain, as an induced subgraph, a special double-connected graph, which is either one of the following graphs or th...

Let G be a simple graph on n vertices having edge-connectivity 1(' (G) > O. We say G is k-critical if 1(' (G) :::: k and 1(1 (G-e) < k for every edge e of G. We denote by ~(n,k) the set of all k-critical graphs on n vertices. In this paper we prove that the maximum number of edges of a graph G in ~(n,k) to be: k(n-k) if n ~ 3k; and L ~ (n+k)2 J. if k + 1 s n < 3k. Further, we characterise the e...

An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3–connected. Tutte’s Wheels Theorem proves that the only simple 3–connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple 3-connected graphs with at most two non-essential edges. ...

‎let g=(v,e) be a simple connected graph with vertex set v and edge set e. the first, second and third zagreb indices of g are respectivly defined by: $m_1(g)=sum_{uin v} d(u)^2, hspace {.1 cm} m_2(g)=sum_{uvin e} d(u).d(v)$ and $ m_3(g)=sum_{uvin e}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in g and uv is an edge of g connecting the vertices u and v. recently, the first and second m...

The class of star graphs is a popular topology for interconnection networks. However it has certain deficiencies. A class of generalization of star graphs called (n, k)-star graphs was introduced in [12] to address these issues. In this paper we will consider the vertex-connectivity of the directed (n, k)-star graph, −−→ Sn,k, given in [8], and show that it is maximally connected.

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G−F has no isolated vertex. The restricted edge connectivity λ′ is theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG is called ...

We describe a method for drawing graph-enhanced Euler diagrams using a three stage method. The first stage is to lay out the underlying Euler diagram using a multicriteria optimizing system. The second stage is to find suitable locations for nodes in the zones of the Euler diagram using a force based method. The third stage is to minimize edge crossings and total edge length by swapping the loc...

Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely Kr-saturated graphs and find the list of uniquely K4-saturated graphs on at most 12 vertices, supporting curre...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using 5 colors. This result is tight since there are...